Geometry & Trigonometry

Coordinate geometry (analytic geometry) places geometric shapes on the xy-plane so we can study them with algebra.

Key Formulas

• Distance: d = √((x₂-x₁)² + (y₂-y₁)²). This is just the Pythagorean theorem applied to the horizontal and vertical differences.
• Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2). Average the coordinates.
• Gradient (Slope): m = (y₂-y₁)/(x₂-x₁). Rise over run.

Chinese textbooks call the gradient 斜率 and write it k, with k = tan α, where α is the 倾斜角 (inclination angle, 0° ≤ α < 180°). A vertical line (α = 90°) has no slope. Lines are written y = kx + b (b = 截距, the y-intercept).

Gradient Relationships

• Parallel lines: same gradient → m₁ = m₂.
• Perpendicular lines: gradients are negative reciprocals → m₁ · m₂ = -1.

Collinear Points
Three points are collinear (on the same line) if the gradient between any two pairs is equal.

Section Formula
The point dividing the segment from (x₁,y₁) to (x₂,y₂) in the ratio m:n is: ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).

These basics underpin every geometry question. The distance formula alone can determine whether a triangle is isosceles, right-angled, or equilateral.

Locus Problems
A locus is the set of all points that satisfy a stated geometric condition, and most locus questions are really conic questions in disguise. Learn the four standard conditions and the answer follows immediately:

• Equidistant from two fixed points → the perpendicular bisector of the segment joining them. Points equidistant from (−2, 0) and (4, 0) form the vertical line x = 1.
• A fixed distance from one fixed point → a circle.
• Equidistant from a point and a line (the point not on the line) → a parabola, with the point as focus and the line as directrix.
• The sum of the distances from two fixed points is constant and greater than the distance between them → an ellipse with those points as foci. The absolute difference being a non-zero constant less than the distance between them gives a hyperbola instead.

Check the condition. If the sum equals the distance between the points, the locus is only the segment joining them: |PF₁| + |PF₂| = 6 with F₁(−3, 0), F₂(3, 0) is the segment from (−3, 0) to (3, 0), not an ellipse. If the point lies on the line, "equidistant from point and line" gives the perpendicular line through the point, not a parabola.

So a point whose distances from (3, 0) and (−3, 0) always add to 10 traces an ellipse: 2a = 10 gives a = 5, and c = 3 gives b² = a² − c² = 16, so the locus is x²/25 + y²/16 = 1. When no standard condition fits, fall back on the method itself: write the condition as an equation in the distances from a general point (x, y), square to clear the roots, and simplify.

The Centroid of a Triangle
The three medians of a triangle — each joining a vertex to the midpoint of the opposite side — always meet at one point, the centroid, and in coordinates it is simply the average of the three vertices:

G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)

For vertices (0, 0), (6, 0) and (3, 9) the centroid is ((0+6+3)/3, (0+0+9)/3) = (3, 3). The centroid is the triangle’s balance point, and it divides each median in the ratio 2 : 1 from the vertex. The same averaging works with position vectors, so the centroid of A, B and C has position vector (a + b + c)/3.